Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y) = 2x(1 – y); (b) u(x, y) = 2x - x3 + 3xy2;

Q: Find Vf at the given point. f(x, y, z) = 32³ – 2(x² + y²) z + tan '(4xz), (0, 2, 3) Vf\(0,2,3) f(x,…

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Q: Let u(x, y) = e=² -y², cos 2xy. (a) Verify that u(x, y) is harmonic in R?. (b) Find its harmonic…

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Q: Find Wr 0 if W = f(x,y) x=r cos 0 and y = r sin 0, then prove that r Wr 0 =yfx + xfy + xy(fyy-fxx) +…

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Q: For F(x, y, z) = x²z² i+ sinx z²j+zx²y*k, find div F and curl F.

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Q: Integrate counterclockwise dz by Cauchy's formula z2 – 16 counterclockwise around the circle C : |z…

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Q: what is the area of the rotational surface formed by rotating the part of the function y= x^3/3…

A: We apply the above formula to solve this

Q: . Verify that u(x, y) = xy+7x +e° onjugates v(x, y) and find an expression for f(2) = f(x+iy) = u(x,…

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Q: 3. Verify that u(x, y) = x² + 4.x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: See the attachment

Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: w In(x2 + y2 +z?), x = ue" sin u, y = ue" cos u and z ue", using chain rule at the point (u, v) =…

A: Given w=lnx2+y2+z2x=uevsinuy=uevcosuz=uev

Q: Find the solution u(t, x) to the system Jau +30₂u+u²=0 for t≥0, 20, u(t = 0, x) = cos(x). Start by…

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Q: - Find div F and curl F if F(x, y, z) = 10y¹z7i – 17xªzºj – 3xyễk. div F= curl F=

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

Q: tu+v dt In(t) f(u, v) = u-1 fy Find using the chain rule u = x +y³, v= 3y- 3xs

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Q: dz (a) Use an appropriate form of the chain rule to find dx dz in terms of y in terms of x and if z…

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Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

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Q: 1. Give an equation of the tangent plane to the surface z = In(4y – x) at the point (3,1,0). 2. Use…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 2. Find the directional drevative of f(x, y,z) xsinyz at the point (1,3,0) in the direction of v = i…

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Q: A) Show that from the lower relations near the point (0,0,0, 0) u can be written as a derivative…

A: Given relations: vcos(y)+xcos(u)=0 ucos(x)+ycos(v)=0 To find the second order Taylor polynomial of u…

Q: Let z = g (x, y) = x³ + xy³ + y4 - 6x - 2y + 4, where x = r cos (0) and y дz Use the chain rule to…

A: We will use chain rule to find the derivative

Q: Find div F and curl F if F(x, y, z) = 13y'z4i – 16x4z10j – 3xy'k. - - div F= curl F=

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Q: Show that u(x, y) is harmonic in some domain. u(x, y) = 2x – x³ + 3xy?

A: since Uxx+Uyy=0, then U is harmonic

Q: Find the linearization L(x,y) of the function f(x,y) = e (X cos (5y) at the points (0,0) and 0,- 2

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Q: Find and ди ди using the chain rule. w(x, y) = 14x²y², x = e" cos v, (u, υ) ди (u, υ)

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Q: Solve by theorems of linearity and translation. f(t) = cos t + sin t

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Q: 3. Find the circulation of F 2xi + 2zj +2yk around closed contour %3D

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Q: Let r(t) = (sin(t), cos(t), 5 sin(t) + 4 cos(2t)). Find the projection of r(t) onto the xz-plane for…

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Q: Verify that u(x, y) = x² + 4x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: The solution is given as

Q: (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by…

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Q: (b) Use Leibniz's product rule to derive the quotient rule y du- v dy d YY

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Q: ▼ 3.1. Use Cauchy-Riemann equations to prove that ƒ(z) = z² is differentiable at z = zo E C.

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Q: 2. Find an equation for the plane tangent to the surface 3x2 - 4xy+ z2 = 0 at the point (1, 1, 1).…

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Q: Find fæ, fy, and fz. f(z, y, z) = z In(a"y cos(5z) 4.3 = z In (x*y° cos(52)) fr = fy = f: =

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Q: 2- Show that the function U(x,y)= 3x²y-y² + y is Harmonic and then find it's Harmonic conjugate V(x,…

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Q: Find fa, fy, and fz. f (x, y, z) = z In(xʻy° cos(52)) 4,3 COS x*Y fn = fy = fz =

A: Please refer the attached image for complete solution.

Q: 1. (a) Let /(r,y)- cos(z + 2y). Compte the directional derivative Da/(r.y), where u- (b) Compute ff,…

A: Given that, In part (a), f(x, y) = cos(x+2y) u=(x,y)=-12,12 We have to find the directional…

Q: Show that U (x, y) = e"cos v,V(x, y) = e"sin v are conjugate harmonic of each other if f = u + iv is…

A: Solution given below

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

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Q: 2. (a) Let z ry, r =r cost and y=r sin t. Compute by using the chain rule and express your answer in…

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Q: (a) Consider the equation e'ux+2ayuy +2e*uy-sin(y) u =0, where a is a real constant. Determine the…

A: We will use basic knowledge of PDEs to answer this question.

Q: Show that U(x, y) = e"cos v,V(x, y) = e"sin harmonic of each other if f = u + iv is analytic. v are…

A: Given: The function f=u+iv is analytic.   To show: Ux,y=eucosv and Vx,y=eusinv are conjugate…

Q: Prove that arctan x + arctan y = arctan(x + y) / (1 − xy),  xy ≠ 1.                   Use the…

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Q: Show that u (x, y) is harmonic in some domain and |find a harmonic conjugate v(x, y) when y (a) u(x,…

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Q: The imaginary part of z=x+iy is x

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Q: Let w = f(z) = z + 3z=u+iv where, z is a complex variable such that z=x+ iy then, the value of v at…

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Q: 2. Find the directional drevative of f(x, y,z) xsinyz at the point (1,3,0) in the direction of i =i+…

A: Use projection formula

Q: Find the directional derivate of f (x, y) = tan (3xy) at the point (1,2) in the direction of 3i - j.

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Q: Find the analytic functiopn f (z) = u (x,y) + i ( x, y ) such that u ( x, y)=  4xy3-4x3y+x.

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